Robust H∞ load frequency control of delayed multi-area power system with stochastic disturbances☆
Introduction
Frequency stability is an important factor in the safe and stable operation of power system, which can reflect the basic status of active power balance between supply and demand. Frequency fluctuations usually bring some serious consequences to power plants and safe operation of power systems, as well as consumers, for instance, it makes the generator sets and auxiliary power machines deviate from operating condition, thus reduces the mechanical efficiency and makes power plant operation deviate from economic efficiency, finally affects the entire economic operation of power grid [1], [2]. Besides, when frequency is too low, it even threatens the safe operation of entire network. Therefore, frequency analysis and control of power system has been assumed to be the non-ignorable element in the safe operation of power system, especially for the modern wide-area power system. Load frequency control (LFC) has become the main way to balance the power and frequency of interconnected power grid, thus the quality of power grid can be directly influenced by the control performance [3], over the past few decades, LFC of power system has attracted wide considerations, and lots of related results have been developed [4], [5].
With the growing scale of modern power grid, power system is gradually evolved from the traditional single area system to the current complex multi-area interconnected systems, which makes it inevitable to keep the fundamental constant of power system. Generally speaking, power system can maintain the frequency in the allowable deviation range via the primary and secondary frequency regulations [1]. However, due to the information exchange between different regions in multi-area power system and the constantly changing load, robust issues about LFC of power system have attracted a widespread concerns in the past few years [6], [7]. In [8], a robust decentralized load frequency controller was proposed for interconnected power systems. In [9], based on the unknown input observers, a robust sensor and actuator fault detection and isolation method was proposed for LFC of interconnected power systems. In [10], a coordination of V2G control and conventional frequency controller was designed for robust LFC of smart grid with the penetration of wind farms. For the other related results, refer to [11] and the references therein.
On the other hand, time delay caused by signal transmission between different control areas in multi-area power systems cannot be ignored, which would degrade the dynamic performance of LFC, and cause instability in the worst case. In [12], the authors discussed the impact of time delay on LFC of microgrid, a new method calculating the critical stable delay margin was developed. In [13], by using linear matrix inequality (LMI) approach, a robust decentralized control approach was proposed for the design of LFC with respect to communication delays and failures. In [14], a robust two-term controller designing strategy was proposed for interconnected power systems with communication delay. In [15], based on Lyapunov stability theory, the authors investigated the delay-dependent stability of LFC scheme with constant and time-varying delays. In [16], an improved delay-dependent stability condition was developed for LFC emphasizing on multi-area and deregulated environment, which could improve calculation accuracy and reduce computation time. However, it should be pointed out that most of the aforementioned results only considered time delay and its effects in signal area, the transmission delays between different control areas in multi-area power systems have not been fully discussed, which plays an important role in analyzing LFC of multi-area power system, especially under the wide area control environment.
Meanwhile, any physical systems, such as energy system [17], [18], neural networks [19], [20], [21], and biological system [22], [23], [24], are always perturbed by stochastic disturbances from the environment. Power system cannot be also exempted, randomness and uncertainties in power system, such as stochastic loads [25], inherent randomness in wind power generation [26], [27], random vibration of original motivation [1], and random small oscillation of power angle [2], will inevitably affect the safe and stable operation of power system. In [28], both load and wind power production were modeled by stochastic differential equations to address the balance management problem of power system in an hourly time frame. In [29], the authors considered the small signal stability of power system under Gauss random excitations. In [30], a systematic stochastic modeling approach was proposed for power system, and numerical stability was discussed in detail. In [31], the authors considered robust stochastic stability of power system with time-varying delay under Gaussian random perturbations. However, to the best of the authors׳ knowledge, there are few results considering robust LFC of delayed multi-area power system with stochastic perturbations, not to mention the delayed feedback controller design under wide area environment. Based on the above discussions, in this paper, considering stochastic disturbances induced by the integration of large number of renewable energy resources into power grid, the robust LFC problem of delayed multi-area power system with parametric uncertainties and stochastic disturbances is investigated, where the state feedback controller and delayed state feedback controller will be designed to realize the robustly LFC of interconnected power systems with performance, respectively.
The rest of this paper is arranged as follows. In Section 2, the dynamic model of delayed multi-area power system with stochastic disturbances is presented, and some useful lemmas needed in the proof are also provided. In Section 3, by using two effective control strategies, the main results are developed. In Section 4, an illustrative example is provided to illustrate the usefulness and effectiveness of the developed results. At last, this paper completes with a conclusion.
Notations: The notations used in this paper are fairly standard. Rn denotes the n-dimensional Euclidean space, when the size is not relevant or can be determined from the context, the subscripts n or m×n will be omitted. AT denotes the transpose of matrix A. means A is a positive definite matrix. denotes a block-diagonal matrix. The notation ⁎ is used to indicate the terms that can be induced by symmetry. A−1 denotes the inverse of matrix A.
Section snippets
Model description
The main aim of the paper is to investigate the robust LFC problem of delayed multi-area power system with stochastic disturbances, for convenience, a two-area interconnected power system with transmission delays is utilized to illustrate the LFC problem, the block diagram of the LFC system can be found in Fig. 1.
It follows from Fig. 1 that the dynamic model of delayed multi-area power system can be formulated as
Controller designing strategies
In this section, two different controller designing strategies are proposed to design the robust load frequency controller for multi-area power system (3) with stochastic disturbances.
Simulation results
In this section, simulation results based on the two-area interconnected power system are provided to demonstrate the usefulness and effectiveness of the proposed controller designing strategies, where Area 1 is modeled with two generators and Area 2 is modeled with four generators [14], some corresponding parameters are given in Table 1.
For the considered LFC system (2), without loss of generality, time delays in two control areas are manually set as and . If there are no external
Conclusions
In this paper, the robust LFC problem of delayed multi-area power system with stochastic disturbances has been investigated. Considering the transmission delay between different control areas, state feedback control and delayed feedback control approaches were proposed, based on Lyapunov stability theory, then several less conservative stability and robust stability results have been developed, respectively, by which robust stability of LFC system with parametric uncertainties and stochastic
Yonghui Sun received the M.S. degree in applied mathematics from Southeast University, Nanjing, China, in 2007, and the Ph.D. degree in control theory and application from City University of Hong Kong, Hong Kong, in 2010.
He is currently a Professor in the College of Energy and Electrical Engineering, Hohai University. He is an active reviewer for many international journals. His research interests include analysis and control of power systems, stochastic control, complex networks, systems
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Yonghui Sun received the M.S. degree in applied mathematics from Southeast University, Nanjing, China, in 2007, and the Ph.D. degree in control theory and application from City University of Hong Kong, Hong Kong, in 2010.
He is currently a Professor in the College of Energy and Electrical Engineering, Hohai University. He is an active reviewer for many international journals. His research interests include analysis and control of power systems, stochastic control, complex networks, systems biology, and fuzzy modeling and control.
Ning Li received the B.S. degree in automation from Hohai University, Nanjing, China, in 2015, where he is currently working toward his Master degree in Control Theory and Applications in the same university. His research interests are distributed control of power systems, load frequency control.
Xuemao Zhao received the bachelor's degree in electrical engineering and the automatization from Hefei University, China, in 2013, and currently he is working toward his Master degree in Hohai University, China. His research interests include load frequency control, stability analysis of power system, etc.
Zhinong Wei received the B.S. degree from Hefei University of Technology, Hefei, China, in 1984, the M.S. degree from Southeast University, Nanjing, China, in 1987, and the Ph.D. degree from Hohai University, Nanjing, China, in 2004.
He is now a Professor of electrical engineering with the College of Energy and Electrical Engineering, Hohai University, Nanjing, China. His research interests include state estimation, voltage stability, smart distribution systems, optimization and planning, load forecasting, and integration of distributed generation into electric power systems.
Guoqiang Sun received the B.S., M.S., and Ph.D. degrees in electrical engineering from Hohai University, Nanjing, China, in 2001, 2005, and 2010, respectively.
He is now an Associate Professor with the College of Energy and Electrical Engineering, Hohai University, Nanjing, China. His research interests are power system analysis and its control. His research interests are power system analysis and its control.
Can Huang received the B.S.E.E degree from Hohai University, Nanjing, China, in 2008, and the M.S.E.E. degree from Southeast University, Nanjing, China, in 2011. From 2011 to 2012, he was with the State Grid Electric Power Research Institute (NARI Group Corporation), Nanjing, China. He is currently a Ph.D. candidate in electrical engineering at the University of Tennessee, Knoxville, TN, USA.
His current research interests include renewable energy, power system operation and control, and IT applications in power system measurement, protection, and communication.